Tight bounds on the randomized communication complexity of symmetric XOR functions in one-way and SMP models

We study the communication complexity of symmetric XOR functions, namely functions f : {0, 1}×{0, 1} → {0, 1} that can be formulated as f(x, y) = D(|x⊕ y|) for some predicate D : {0, 1, ..., n} → {0, 1}, where |x⊕ y| is the Hamming weight of the bitwise XOR of x and y. We give a public-coin randomized protocol in the Simultaneous Message Passing (SMP) model, with the communication cost matching the known lower bound for the quantum and two-way model up to a logarithm factor. As a corollary, this closes a quadratic gap between quantum lower bound and randomized upper bound for the one-way model, answering an open question raised in Shi and Zhang [SZ09].